A critical point is any input to a function where the derivative of the function is or .

Critical points are useful in finding local and absolute extrema using the first derivative and second derivative tests.

Examples

Non-Existent Derivative

Given the function find critical points

is shown in blue and the derivative of is also shown in green.

There is a critical point at . This is shown in the graph as the derivative of shown in green does not exist at .

Zero Derivative (Horizontal Slope)

Given the function find critical points

is shown in blue and the derivative of is also shown in green.

There is a critical point at . This is shown in the graph as the derivative of crosses the x-axis at .